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M32-13

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M32-13  [#permalink]

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New post 17 Jul 2017, 09:20
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If \(n = x^5*y^7\), where \(x\) and \(y\) are positive integers greater than 1, then how many positive divisors does \(n\) have?



(1) \(x\) does not have a factor \(p\) such that \(1 < p < x\) and \(y\) does not have a factor \(q\) such that \(1 < q < y\).

(2) \(n\) has only two prime factors.

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Re M32-13  [#permalink]

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New post 17 Jul 2017, 09:20
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Official Solution:


If \(n = x^5*y^7\), where \(x\) and \(y\) are positive integers greater than 1, then how many positive divisors does \(n\) have?

(1) \(x\) does not have a factor \(p\) such that \(1 < p < x\) and \(y\) does not have a factor \(q\) such that \(1 < q < y\). This statement implies that both \(x\) and \(y\) are primes (a prime number does not have a factor which is greater than 1 and less than itself, it has only two factors 1 and itself). Now, if \(x\) and \(y\) are different primes, then the number of factors of \(n\) will be \((5+1)(7+1)=48\) but if \(x=y\), then \(n = x^5*y^7=x^{12}\) and it will have 13 factors. Not sufficient.

(2) \(n\) has only two prime factors. If those primes are \(x\) and \(y\), then the number of factors of \(n\) will be \((5+1)(7+1)=48\) but if, say \(x=2*3=6\) and \(y=2\), then \(n = x^5*y^7=2^{12}*3^5\) and it will have \((12+1)(5+1)=78\) factors. Not sufficient.

(1)+(2) Since from (2) \(n\) has only two prime factors, then from (1) it follows that \(x\) and \(y\) are different primes so the number of factors of \(n\) will be \((5+1)(7+1)=48\). Sufficient.


Answer: C
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Re: M32-13  [#permalink]

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New post 18 Jul 2017, 23:20
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What about when x is exactly 1? That satisfies statement one (x has no factor greater than 1 and less than itself), but would lead to answer E eventually.

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Re: M32-13  [#permalink]

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Re: M32-13  [#permalink]

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New post 23 Feb 2018, 17:02
Statement 1 raises the fact that x and y could be different or the same. statement 2 says that n has two prime factors. when the statements are combined together it is possible for x is equal to 6 and y is 2. Here the n has two prime factors and they are different. Also x could equal 2 and y equal 3. Both scenarios give us different factors. The answer should be E.
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Re: M32-13  [#permalink]

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New post 23 Feb 2018, 23:31
etibamanya wrote:
Statement 1 raises the fact that x and y could be different or the same. statement 2 says that n has two prime factors. when the statements are combined together it is possible for x is equal to 6 and y is 2. Here the n has two prime factors and they are different. Also x could equal 2 and y equal 3. Both scenarios give us different factors. The answer should be E.


The answer should be and is C, not E.

(1) means that x and y are primes, so x cannot be 6. 6 does have factors which are more than 1 and less than 6 itself: 2 and 3.
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Re: M32-13  [#permalink]

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New post 24 Feb 2018, 07:11
Is there a place where these questions are just listed? I notice that often the title of the post is something like "Mxy-zw" - are they being taken from a book or something? And how do I get my hands on it? :-)

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Re: M32-13  [#permalink]

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New post 01 Sep 2018, 08:42
My initial instinct in seeing two variables x and y is that they are different values. Based on this answer explanation I should completely scrub this line of thinking from my mind for the GMAT, right?
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Re: M32-13  [#permalink]

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New post 02 Sep 2018, 02:02
pzone123 wrote:
My initial instinct in seeing two variables x and y is that they are different values. Based on this answer explanation I should completely scrub this line of thinking from my mind for the GMAT, right?


Right. Unless it is explicitly stated otherwise, different variables CAN represent the same number.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: M32-13 &nbs [#permalink] 02 Sep 2018, 02:02
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